$b^k$-Symplectic Manifolds and $[Q,R]=0$

Abstract

We study the geometric quantization of $b^k$-symplectic manifolds using the integrability of Lie algebroids. Using a groupoid index, we define a quantization for $b^k$-symplectic manifolds whose singular locus is a normal crossing divisor and which carry a Hamiltonian action of a compact connected Lie group, generalizing Guillemin--Miranda--Weitsman in two directions. We show this quantization is the index of a Spin$^c$-Dirac operator, answering a question of theirs; in particular, it is a finite-dimensional virtual representation for every $k$, whereas their formal quantization is infinite-dimensional when the modular degrees are even. Finally, we prove that quantization commutes with reduction when the modular degrees are odd. This hypothesis is necessary rather than technical: a finite-dimensional index cannot agree with an infinite-dimensional formal quantization. We exhibit an explicit example where $[Q,R]=0$ fails for even modular degree.